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The formula for the sampling interval in systematic random sampling is:
\[k = \frac{N}{n}\]
Where:
- \(N\) = population size = 400
- \(n\) = desired sample size = 50
The nurse randomly selects a starting point between 1 and 8, then selects every 8th patient thereafter (e.g., patient 5, 13, 21, 29...). This method ensures equal probability of selection and is more practical than simple random sampling for large populations.
1. Calculate Refractive Index ($\mu$):
$\mu = \frac{c}{v} = \frac{3 \times 10^8}{2 \times 10^8} = 1.5$.
2. Determine Radius of Curvature ($R$):
For a lens with radius of aperture $r = 3 \text{ cm}$ and thickness $t = 0.3 \text{ cm}$, the radius of curvature is given by $R \approx \frac{r^2}{2t}$ (for small thickness).
$R = \frac{3^2}{2 \times 0.3} = \frac{9}{0.6} = 15 \text{ cm}$.
3. Lens Maker's Formula:
$\frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)$. For a plano-convex lens, $R_1 = R$ and $R_2 = \infty$.
$\frac{1}{f} = (1.5 - 1) \left( \frac{1}{15} \right) = 0.5 \times \frac{1}{15} = \frac{1}{30}$.
Thus, $f = 30 \text{ cm}$.
In a gaseous mixture, the ratio of masses of oxygen to nitrogen is 1 : 4. What is the ratio of the number of molecules of O$_2$ to N$_2$?
Let mass of O$_2 = 1$ g and mass of N$_2 = 4$ g.
Moles of O$_2 = \dfrac{1}{32}$ ; Moles of N$_2 = \dfrac{4}{28} = \dfrac{1}{7}$
Ratio $= \dfrac{1/32}{1/7} = \dfrac{7}{32}$
So the ratio of molecules O$_2$ : N$_2 = \mathbf{7 : 32}$
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